Stock Returns Asymmetry: Beyond Skewness

In this paper, we propose two asymmetry measures of stock returns. In contrast to the usual measure, skewness, ours are based on the tail distribution of the data instead of just the third moment. While it is inconclusive with the skewness, we find that, with our new measures, greater upside asymmetries imply lower average returns in the cross section, consistent with theoretical models such as Barberis and Huang (2008).

Stock Returns Asymmetry: Beyond Skewness – Introduction

In theory, Tversky and Kahneman (1992), Polkovnichenko (2005), and Barberis and Huang (2008) show that greater upside asymmetry is associated with lower expected return. Empirically, using skewness, the most popular measure of asymmetry, Harvey and Siddique (2000), Zhang (2005), Smith (2007), Boyer et al. (2010), and Kumar (2009) find empirical evidence supporting the theory. However, Bali et al. (2011) find that skewness is not statistically significant in explaining the expected returns in a more general set-up. More recently, An et al. (2015) find that the correlation between skewness and expected return depends on capital gains overhang (CGO). In short, the evidence on the ability of skewness in capturing asymmetry to explain the cross-section stock returns is mixed and inconclusive.

In this paper, we propose two distribution-based measures of asymmetry. We argue that skewness, as a measure of asymmetry, is limited because how two distributions can have the same skewness while quite different in asymmetry. Intuitively, asymmetry reflects a characteristic of the entire distribution, but skewness consists of only the third moment. Therefore, even if the skewness is inconclusive in explain asset returns, it does not mean asymmetry matters any less. This clearly comes down to how we can measures asymmetry adequately. Our first measure of asymmetry is a simple difference between the upside probability and downside probability, which captures the degree of upside asymmetry based on probabilities. The second measure is a modified entropy measure, modified from Racine and Maasoumi (2007), that assesses asymmetry based on integrated density difference. Statistically, we show via simulations that our distribution-based measures can capture asymmetry more accurately than skewness, the third moment only measure.

Empirically, we examine the explanatory power of both skewness and our new measures in the cross-section of stock returns, and find that our measures explain well the returns and skewness does not. We conduct our analysis with two approaches. In the first approach, we study their performances in explaining the returns using Fama and MacBeth (1973) regressions. Using data from January 1962 to December 2013, we find that there is no apparent relation between the skewness and the cross-sectional average returns, which is consistent with the findings by Bali et al. (2011). In contrast, based on our new measures, we find that asymmetry does matter in explaining the cross-sectional variation of stock returns. The greater the upside tail asymmetry, the lower the average returns in the cross-section. In the second approach, we sort stocks into decile portfolios of high and low asymmetry with respect to skewness or to our new asymmetry measures. We find that while high skewness portfolios do not necessarily imply low returns, high upside asymmetries based on the new measures do associate with low returns.

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